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[Calculus](https://www.mathworks.com/help/symbolic/calculus.html?s_tid=CRUX_lftnav) - laplace - On this page - [Syntax](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266166) - [Description](https://www.mathworks.com/help/symbolic/sym.laplace.html#description) - [Examples](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266245) - [Laplace Transform of Symbolic Expression](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_ab29ba7c-c6bc-4427-9590-095873d039d5) - [Specify Independent Variable and Transformation Variable](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_f371bf19-fe52-4f7e-92e2-32464f819335) - [Laplace Transforms of Dirac and Heaviside Functions](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_4f3dac42-6a50-45c6-87b6-940cd6060258) - [Relation Between Laplace Transform of Function and Its Derivative](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_87a5aa1a-bf7e-4612-810f-d3f98513a212) - [Laplace Transform of Array Inputs](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_580c9f73-b9db-4803-939f-a24742e95023) - [Laplace Transform of Symbolic Function](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_81e6639d-3c84-4c84-9426-2e28a1cb734d) - [If Laplace Transform Cannot Be Found](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_bfa6d5f7-833b-4424-8a5f-0c84c4731631) - [Input Arguments](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266749) - [f](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266753) - [var](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266776) - [transVar](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266810) - [More About](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266869) - [Laplace Transform](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_d8b5d523-43d3-4d19-9285-2d39828e1356) - [Tips](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266854) - [Algorithms](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_9b0d4f01-b0d9-4b0a-9141-e44a7335cbd4) - [Version History](https://www.mathworks.com/help/symbolic/sym.laplace.html#btfobhn_vh) - [See Also](https://www.mathworks.com/help/symbolic/sym.laplace.html#btfobhn_seealso) - [Documentation](https://www.mathworks.com/help/symbolic/calculus.html?s_tid=CRUX_topnav) - [Examples](https://www.mathworks.com/help/symbolic/examples.html?s_tid=CRUX_topnav&category=calculus) - [Functions](https://www.mathworks.com/help/symbolic/referencelist.html?type=function&s_tid=CRUX_topnav&category=calculus) - [Videos](https://www.mathworks.com/support/search.html?fq%5B%5D=asset_type_name:video&fq%5B%5D=category:symbolic/calculus&page=1&s_tid=CRUX_topnav) - [Answers](https://www.mathworks.com/support/search.html?fq%5B%5D=asset_type_name:answer&fq%5B%5D=category:symbolic/calculus&page=1&s_tid=CRUX_topnav) Main Content # laplace Laplace transform [collapse all in page]() ## Syntax `F = laplace(f)` `F = laplace(f,transVar)` `F = laplace(f,var,transVar)` ## Description `F = laplace(f)` returns the [Laplace Transform](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_d8b5d523-43d3-4d19-9285-2d39828e1356) of `f`. By default, the independent variable is `t` and the transformation variable is `s`. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_ab29ba7c-c6bc-4427-9590-095873d039d5) `F = laplace(f,transVar)` uses the transformation variable `transVar` instead of `s`. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_f371bf19-fe52-4f7e-92e2-32464f819335) `F = laplace(f,var,transVar)` uses the independent variable `var` and the transformation variable `transVar` instead of `t` and `s`, respectively. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_4f3dac42-6a50-45c6-87b6-940cd6060258) ## Examples [collapse all]() ### Laplace Transform of Symbolic Expression [Open Live Script](matlab:openExample\('symbolic/LaplaceTransformOfSymbolicExpressionExample'\)) Compute the Laplace transform of `1/sqrt(x)`. By default, the transform is in terms of `s`. ``` syms x y f = 1/sqrt(x); F = laplace(f) ``` ``` F =  π s ``` ### Specify Independent Variable and Transformation Variable [Open Live Script](matlab:openExample\('symbolic/SpecifyIndependentVariableAndTransformationVariableExample'\)) Compute the Laplace transform of `exp(-a*t)`. By default, the independent variable is `t`, and the transformation variable is `s`. ``` syms a t y f = exp(-a*t); F = laplace(f) ``` ``` F =  1 a + s ``` Specify the transformation variable as `y`. If you specify only one variable, that variable is the transformation variable. The independent variable is still `t`. ``` F = laplace(f,y) ``` ``` F =  1 a + y ``` Specify both the independent and transformation variables as `a` and `y` in the second and third arguments, respectively. ``` F = laplace(f,a,y) ``` ``` F =  1 t + y ``` ### Laplace Transforms of Dirac and Heaviside Functions [Open Live Script](matlab:openExample\('symbolic/LaplaceTransformsOfDiracAndHeavisideFunctionsExample'\)) Compute the Laplace transforms of the Dirac and Heaviside functions. ``` syms t s syms a positive F = laplace(dirac(t-a),t,s) ``` ``` F = e - a s ``` ``` F = laplace(heaviside(t-a),t,s) ``` ``` F =  e - a s s ``` ### Relation Between Laplace Transform of Function and Its Derivative [Open Live Script](matlab:openExample\('symbolic/LaplaceTransformOfFunctionAndItsDerivativeExample'\)) Show that the Laplace transform of the derivative of a function is expressed in terms of the Laplace transform of the function itself. ``` syms f(t) s Df = diff(f(t),t); F = laplace(Df,t,s) ``` ``` F = s laplace ( f ( t ) , t , s ) - f ( 0 ) ``` ### Laplace Transform of Array Inputs [Open Live Script](matlab:openExample\('symbolic/LaplaceTransformOfArrayInputsExample'\)) Find the Laplace transform of the matrix `M`. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, `laplace` acts on them element-wise. ``` syms a b c d w x y z M = [exp(x) 1; sin(y) 1i*z]; vars = [w x; y z]; transVars = [a b; c d]; F = laplace(M,vars,transVars) ``` ``` F =  ( e x a 1 b 1 c 2 + 1 i d 2 ) ``` If `laplace` is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size. ``` F = laplace(x,vars,transVars) ``` ``` F =  ( x a 1 b 2 x c x d ) ``` ### Laplace Transform of Symbolic Function [Open Live Script](matlab:openExample\('symbolic/LaplaceTransformOfSymbolicFunctionExample'\)) Compute the Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar. ``` syms f1(x) f2(x) a b f1(x) = exp(x); f2(x) = x; F = laplace([f1 f2],x,[a b]) ``` ``` F =  ( 1 a - 1 1 b 2 ) ``` ### If Laplace Transform Cannot Be Found [Open Live Script](matlab:openExample\('symbolic/IfLaplaceTransformCannotBeFoundExample'\)) If `laplace` cannot transform the input then it returns an unevaluated call. ``` syms f(t) s f(t) = 1/t; F(s) = laplace(f,t,s) ``` ``` F(s) =  laplace ( 1 t , t , s ) ``` Return the original expression by using `ilaplace`. ``` f(t) = ilaplace(F,s,t) ``` ``` f(t) =  1 t ``` ## Input Arguments [collapse all]() ### `f` — Input symbolic expression \| symbolic function \| symbolic vector \| symbolic matrix Input, specified as a symbolic expression, function, vector, or matrix. ### `var` — Independent variable `t` (default) \| symbolic variable Independent variable, specified as a symbolic variable. This variable is often called the "time variable" or the "space variable." If you do not specify the variable then, by default, `laplace` uses `t`. If [`f`](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266753) does not contain `t`, then `laplace` uses the function `symvar` to determine the independent variable. ### `transVar` — Transformation variable `s` (default) \| `z` \| symbolic variable \| symbolic expression \| symbolic vector \| symbolic matrix Transformation variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable then, by default, `laplace` uses `s`. If `s` is the independent variable of [`f`](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266753), then `laplace` uses `z`. ## More About [collapse all]() ### Laplace Transform The Laplace transform *F*(*s*) of the expression *f*(*t*) with respect to the variable *t* at the point *s* is a unilateral transform defined by F ( s ) \= ∫ 0 – ∞ f ( t ) e − s t d t . ## Tips - If any argument is an array, then `laplace` acts element-wise on all elements of the array. - If the first argument contains a symbolic function, then the second argument must be a scalar. - To compute the inverse Laplace transform, use `ilaplace`. ## Algorithms The Laplace transform is defined as a unilateral or one-sided transform. This definition assumes that the signal *f*(*t*) is only defined for all real numbers *t* ≥ 0, or *f*(*t*) = 0 for *t* \< 0. Therefore, for a generalized signal with *f*(*t*) ≠ 0 for *t* \< 0, the Laplace transform of *f*(*t*) gives the same result as if *f*(*t*) is multiplied by a Heaviside step function. For example, both of these code blocks: ``` syms t; laplace(sin(t)) ``` and ``` syms t; laplace(sin(t)*heaviside(t)) ``` return `1/(s^2 + 1)`. ## Version History **Introduced before R2006a** ## See Also [`fourier`](https://www.mathworks.com/help/symbolic/sym.fourier.html) \| [`ifourier`](https://www.mathworks.com/help/symbolic/sym.ifourier.html) \| [`ilaplace`](https://www.mathworks.com/help/symbolic/sym.ilaplace.html) \| [`iztrans`](https://www.mathworks.com/help/symbolic/sym.iztrans.html) \| [`ztrans`](https://www.mathworks.com/help/symbolic/sym.ztrans.html) ### Topics - [Solve Differential Equations of RLC Circuit Using Laplace Transform](https://www.mathworks.com/help/symbolic/solve-differential-equations-using-laplace-transform.html) Thank you for your feedback\! Why did you choose this rating? 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Laplace transform ## Description `F = laplace(f)` returns the [Laplace Transform](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_d8b5d523-43d3-4d19-9285-2d39828e1356) of `f`. By default, the independent variable is `t` and the transformation variable is `s`. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_ab29ba7c-c6bc-4427-9590-095873d039d5) `F = laplace(f,transVar)` uses the transformation variable `transVar` instead of `s`. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_f371bf19-fe52-4f7e-92e2-32464f819335) `F = laplace(f,var,transVar)` uses the independent variable `var` and the transformation variable `transVar` instead of `t` and `s`, respectively. [example](https://www.mathworks.com/help/symbolic/sym.laplace.html#mw_4f3dac42-6a50-45c6-87b6-940cd6060258) ## Examples collapse all Compute the Laplace transform of `1/sqrt(x)`. By default, the transform is in terms of `s`. ``` syms x y f = 1/sqrt(x); F = laplace(f) ``` ``` F =  π s ``` Compute the Laplace transform of `exp(-a*t)`. By default, the independent variable is `t`, and the transformation variable is `s`. ``` syms a t y f = exp(-a*t); F = laplace(f) ``` ``` F =  1 a + s ``` Specify the transformation variable as `y`. If you specify only one variable, that variable is the transformation variable. The independent variable is still `t`. ``` F = laplace(f,y) ``` ``` F =  1 a + y ``` Specify both the independent and transformation variables as `a` and `y` in the second and third arguments, respectively. ``` F = laplace(f,a,y) ``` ``` F =  1 t + y ``` Compute the Laplace transforms of the Dirac and Heaviside functions. ``` syms t s syms a positive F = laplace(dirac(t-a),t,s) ``` ``` F = e - a s ``` ``` F = laplace(heaviside(t-a),t,s) ``` ``` F =  e - a s s ``` Show that the Laplace transform of the derivative of a function is expressed in terms of the Laplace transform of the function itself. ``` syms f(t) s Df = diff(f(t),t); F = laplace(Df,t,s) ``` ``` F = s laplace ( f ( t ) , t , s ) - f ( 0 ) ``` Find the Laplace transform of the matrix `M`. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, `laplace` acts on them element-wise. ``` syms a b c d w x y z M = [exp(x) 1; sin(y) 1i*z]; vars = [w x; y z]; transVars = [a b; c d]; F = laplace(M,vars,transVars) ``` ``` F =  ( e x a 1 b 1 c 2 + 1 i d 2 ) ``` If `laplace` is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size. ``` F = laplace(x,vars,transVars) ``` ``` F =  ( x a 1 b 2 x c x d ) ``` Compute the Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar. ``` syms f1(x) f2(x) a b f1(x) = exp(x); f2(x) = x; F = laplace([f1 f2],x,[a b]) ``` ``` F =  ( 1 a - 1 1 b 2 ) ``` If `laplace` cannot transform the input then it returns an unevaluated call. ``` syms f(t) s f(t) = 1/t; F(s) = laplace(f,t,s) ``` ``` F(s) =  laplace ( 1 t , t , s ) ``` Return the original expression by using `ilaplace`. ``` f(t) = ilaplace(F,s,t) ``` ``` f(t) =  1 t ``` ## Input Arguments collapse all Input, specified as a symbolic expression, function, vector, or matrix. Independent variable, specified as a symbolic variable. This variable is often called the "time variable" or the "space variable." If you do not specify the variable then, by default, `laplace` uses `t`. If [`f`](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266753) does not contain `t`, then `laplace` uses the function `symvar` to determine the independent variable. Transformation variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable then, by default, `laplace` uses `s`. If `s` is the independent variable of [`f`](https://www.mathworks.com/help/symbolic/sym.laplace.html#d126e266753), then `laplace` uses `z`. ## More About collapse all The Laplace transform *F*(*s*) of the expression *f*(*t*) with respect to the variable *t* at the point *s* is a unilateral transform defined by ## Tips - If any argument is an array, then `laplace` acts element-wise on all elements of the array. - If the first argument contains a symbolic function, then the second argument must be a scalar. - To compute the inverse Laplace transform, use `ilaplace`. ## Algorithms The Laplace transform is defined as a unilateral or one-sided transform. This definition assumes that the signal *f*(*t*) is only defined for all real numbers *t* ≥ 0, or *f*(*t*) = 0 for *t* \< 0. Therefore, for a generalized signal with *f*(*t*) ≠ 0 for *t* \< 0, the Laplace transform of *f*(*t*) gives the same result as if *f*(*t*) is multiplied by a Heaviside step function. For example, both of these code blocks: ``` syms t; laplace(sin(t)) ``` and ``` syms t; laplace(sin(t)*heaviside(t)) ``` return `1/(s^2 + 1)`. ## Version History **Introduced before R2006a** How useful was this information? Unrated1 star2 stars3 stars4 stars5 stars